牛顿法及其下山法+C代码

引用数值分析原文的内容，可以很快的编出牛顿法的代码。牛顿法其原理如下：

从上面理论可以看出，牛顿法就是不断寻找新的点 x(k+1)来逼近目标值，其寻找方法是不断对曲线做切线，并计算“前进”距离：f'(x)=tan( f(x(k)) / (x(k) - x(k+1)))。

下面以书本的例子，编写相应的代码：

下山法是为防止迭代发散而额外加的条件。

其C语言代码如：

#include
#include
#include


#define  E 0.000001
float fun(float x, int chs)
{
float y;


if( 0 == chs )//原函数
y = x*x*x -x -1;


if( 1 == chs )//导数
y = 3*x*x -1;


return y;
}




//牛顿下山法
float newton_downhill_method(float (*fun)(float, int), int iter, float x, float C, float E1, float E2)
{
int k;
int i = 1;
float delta;
float x0, x1;
float f0, df0;
float f1, df1;
float landa = 1;


if ( 0 == C )
{
C = 1;
}


x0 = x;


while( i++ < iter )
{
f0 = fun(x0, 0);
df0 = fun(x0, 1);


x1 = x0 - f0/df0;


f1 = fun(x1, 0);
df1 = fun(x1, 1);


//添加下山法
landa = 1;
while(1)
{
if( fabs(f0) > fabs(f1) )
break;


landa = landa/2;


x1 = x0 - landa*f0/df0;


f1 = fun(x1, 0);


}



if( x1 < C )
delta = fabs(x1 -x0);
else
delta = fabs(x1 -x0)/fabs(x1);


x0 = x1;


printf("%f\n", x0);
if( deltabreak;


if( 0 == df1)
{
printf("Error：df1=0,iterative calculation failure.\n");
break;
}


}


if ( i == iter )
{
printf("Error：The iteration upper limit is reached.\n");
return 0;
}


return x1;


}




//牛顿迭代法（切线法）
float newton_method(float (*fun)(float, int), int iter, float x, float C, float E1, float E2)
{
int i = 1;
float delta;
float x0, x1;
float f0, df0;
float f1, df1;


if ( 0 == C )
{
C = 1;
}


x0 = x;


while( i++ < iter )
{
f0 = fun(x0, 0);
df0 = fun(x0, 1);


x1 = x0 - f0/df0;


f1 = fun(x1, 0);
df1 = fun(x1, 1);


if( x1 < C )
delta = fabs(x1 -x0);
else
delta = fabs(x1 -x0)/fabs(x1);


x0 = x1;


printf("%f\n", x0);


if( deltabreak;


if( 0 == df1)
{
printf("Error：df1=0,iterative calculation failure.\n");
break;
}


}


if ( i == iter )
{
printf("Error：The iteration upper limit is reached.\n");
return 0;
}


return x1;


}


void main()
{
int order;
int iter  = 1000;
float x0  = 0.6;
float C  = 1;


newton_downhill_method(fun, iter, x0, C, E, E);


system("pause");
}

迭代结果：